Basics of maps from tropical cohomology to singular cohomology

Abstract

This is a continuation of our tropical approach to the Hodge conjecture. In this paper, we will construct and study basic properties of maps from tropical cohomology to singular cohomology for smooth complex projective varieties. We will construct two kinds of maps: one is given by limits of differential forms, the other is given by weighted tropicalizations of generic semi-algebaic subsets, which is introduced in this paper. We will show that they are the same under a natural assumption. This paper also contains a new expression of tropical K-groups as logarithmic differentials, which is an analog of Bloch-Kato and Gabber's results of Milnor K-groups in positive characteristic.Comment: 41 pages (Appendix 10 pages) Version 2 and 3: minor change